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Concept
Carathéodory's existence theorem
Summary
In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory.
Consider the differential equation
with initial condition
where the function ƒ is defined on a rectangular domain of the form
Peano's existence theorem states that if ƒ is continuous, then the differential equation has at least one solution in a neighbourhood of the initial condition.
However, it is also possible to consider differential equations with a discontinuous right-hand side, like the equation
where H denotes the Heaviside function defined by
It makes sense to consider the ramp function
as a solution of the differential equation. Strictly speaking though, it does not satisfy the differential equation at , because the function is not differentiable there. This suggests that the idea of a solution be extended to allow for solutions that are not everywhere differentiable, thus motivating the following definition.
A function y is called a solution in the extended sense of the differential equation with initial condition if y is absolutely continuous, y satisfies the differential equation almost everywhere and y satisfies the initial condition. The absolute continuity of y implies that its derivative exists almost everywhere.
Consider the differential equation
with defined on the rectangular domain . If the function satisfies the following three conditions:
is continuous in for each fixed ,
is measurable in for each fixed ,
there is a Lebesgue-integrable function such that for all ,
then the differential equation has a solution in the extended sense in a neighborhood of the initial condition.
A mapping is said to satisfy the Carathéodory conditions on if it fulfills the condition of the theorem.
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